I am having a really hard time figuring out how to convert this circle to polar coordinates, I am to use double integration after converting it. I know that $\theta$ has to be between $0$ and $\frac \pi 2$. I keep trying different things and nothing works. I have been trying to make this shape happen in a graphing calculator, but I cant seem to get a circle that doesnt touch any axis.
If you shift the origin to the point $(5,5)$, then the shaded region becomes the region
outside the circle $x^2+y^2=25$ and inside the square $-5\le x\le 0,\;\; -5\le y\le 0$.
By symmetry, this has the same area as the region outside the circle $x^2+y^2=25$
and inside the square $0\le x\le5,\;\; 0\le y\le5$. Using symmetry again, this gives
$\displaystyle A=2\int_0^{\pi/4}\int_5^{5\sec\theta}r\;dr\;d\theta=2\int_0^{\pi/4}\frac{1}{2}\left(25\sec^2\theta-25\right)d\theta=25\left[\tan\theta-\theta\right]_0^{\pi/4}=25\left(1-\frac{\pi}{4}\right).$